I am trying to model the melting point of a substance at varying pressures (ranging from very small to very very large). All I am trying to do is make an equation that relates melting temperature to pressure, so $T(P)$ is some function. To do this, I am trying to use the Clausius–Clapeyron equation (CC), which states that
$$\frac{\mathrm dP}{\mathrm dT} = \frac{L}{TΔV}.$$
In other words, the slope of the equilibrium line on the phase diagram should decrease as temperature is increased.
However, this is not the case; the curve of the equilibrium line is exponential and the slope $\mathrm dP/\mathrm dT$ increases as $T$ increases. Integrating CC we arrive at a logarithmic function, which again is not what empirical measurements reflect.
As I see it, the empirical results and the equation that is supposed to describe them are mutually exclusive. There is no way to arrive at an exponential curve from a slope that varies with $1/x.$ The CC equation and phase diagrams cannot be both be true at once and it is driving me mad.
Why is this the case? Is the CC equation valid at all because it seems to be totally false? What function do I use to model melting points at different temperatures?
The results that are so stupefying are these:
The shape of the curve is exponential. But, the supposed derivative is $1/T$, in which case the slope of each curve (red and blue here) should flatten as $T$ increases, but it steepens. Also, integrating that supposed derivative gives us $\ln (T)$ which is definitely not the shape of the phase diagram. This discrepancy is true for both the liquid/solid and liquid/gas curves. I hope this clarifies the question!